Problem: Simplify the following expression: $q = \dfrac{20z^2 - 20z}{28z^2 - 32z}$ You can assume $z \neq 0$.
Solution: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $20z^2 - 20z = (2\cdot2\cdot5 \cdot z \cdot z) - (2\cdot2\cdot5 \cdot z)$ The denominator can be factored: $28z^2 - 32z = (2\cdot2\cdot7 \cdot z \cdot z) - (2\cdot2\cdot2\cdot2\cdot2 \cdot z)$ The greatest common factor of all the terms is $4z$ Factoring out $4z$ gives us: $q = \dfrac{(4z)(5z - 5)}{(4z)(7z - 8)}$ Dividing both the numerator and denominator by $4z$ gives: $q = \dfrac{5z - 5}{7z - 8}$